It is known that if \(\left( \frac{1}{p}, \frac{1}{q} \right) \in E_{\mu }\) , then \(\frac{1}{q} \le \frac{1}{p}\)  and
\(\frac{1}{q} \ge \frac{2}{p} -1\)  (cf. [1]}, pp. 33; and [2]}, Theorem 1). Moreover, H1 implies that
\(\frac{1}{q} \ge \frac{1}{p} - \frac{\alpha _1+\alpha _2}{\alpha _1+\alpha _2 + 2m}\)  (cf. [3]}, Proposizione 2.2, pp. 243). In
particular, these facts imply that if \(\left( \frac{1}{p}, \frac{1}{p^{\prime }} \right) \in E_{\mu }\) , then \(\frac{1}{2} \le \frac{1}{p} \le \frac{\alpha _1+\alpha _2+m}{\alpha _1+\alpha _2+2m}\) . On the other hand, by Riesz-Thorin Theorem, \(E_{\mu }\)  is a convex set and, since \(\mu \)  is
finite, the diagonal \(\frac{1}{q} = \frac{1}{p}\)  is contained in \(E_{\mu }\) . Thus, to prove the theorem it is enough to see that
\(\left(\frac{1}{p}, \frac{1}{p^{\prime }} \right) \in E_{\mu }\)  for \(p = \frac{\alpha _1+\alpha _2+2m}{\alpha _1+\alpha _2+m}\) . Consider now, for
\(z \in \mathbb {C}\) , the analytic family of distributions \(J_z\)  defined by (REF ). For \(z\)  in the strip
\(-\frac{\alpha _1+\alpha _2}{m} \le \Re (z) \le 2\) , let \(S_z\)  be the analytic family of operators defined by
\(S_z f = \mu \ast J_z \ast f, \,\,\,\,\,\, f \in \mathcal {S}(\mathbb {R}^{4}).\) 
From Theorem REF , we have \(| \widehat{\mu }(\xi ^{\prime }, \xi ^{\prime \prime })| \le C |\xi ^{\prime \prime }|^{-\frac{\alpha _1+\alpha _2}{m}}\) , and since
\(\widehat{J_z} = 1 \otimes I_{2-z}\)  we obtain that
\(\Vert  S_z \Vert _{L^{2}(\mathbb {R}^{4}) \rightarrow L^{2}(\mathbb {R}^{4})} \le \left\Vert  \widehat{\mu } \widehat{J_z} \right\Vert _{\infty } \le C\left| \frac{2^{1-z}}{\Gamma \left(1-\frac{z}{2}\right)} \right| \,\,\,\,\,\, \text{for} \,\, \Re (z) = -\frac{\alpha _1+\alpha _2}{m},\) 
where \(C\)  is independent of such \(z\) . On the other hand, it is easy to check that
\((\mu \ast J_z)(x^{\prime },x^{\prime \prime }) = \frac{2^{z-1}}{\Gamma \left( \frac{z}{2} \right)} |x^{\prime \prime } - \varphi (x^{\prime })|^{z-2}\)  for \(x^{\prime }, x^{\prime \prime } \in \mathbb {R}^{2}\) 
and so
\(\Vert  S_z \Vert _{L^{1}(\mathbb {R}^{4}) \rightarrow L^{\infty }(\mathbb {R}^{4})} = \Vert  \mu \ast J_z \Vert _{\infty } \le \left| \frac{2^{z-1}}{\Gamma \left( \frac{z}{2} \right)} \right| \,\,\,\,\,\, \text{for} \,\, \Re (z) = 2.\) 
As in Theorem REF , the family \(\lbrace  S_z : -\frac{\alpha _1+\alpha _2}{m} \le \Re (z) \le 2 \rbrace \)  satisfies the hypotheses of the complex interpolation theorem and then, since \(-\frac{\alpha _1+\alpha _2}{m} t+ 2 (1-t)=0\)  for \(t= \frac{2m}{\alpha _1+\alpha _2+2m}\) , \(S_0\)  is a bounded operator from \(L^{p}(\mathbb {R}^{4})\)  into \(L^{p^{\prime }}(\mathbb {R}^{4})\)  for
\(\frac{1}{p} = \frac{t}{2} + (1-t) = \frac{\alpha _1+\alpha _2+m}{\alpha _1+\alpha _2+2m}\) . Finally, being \(T_{\mu }\)  a non-zero multiple of
\(S_0\) , the theorem follows.
